Finite Group Databasebeta

← #26228478

Subgroups — #26228478

Order 12032.

740 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689414C211yesmaximal, characteristic
2#20246134C211yesmaximal, characteristic, complemented
3#20246026C211yesmaximal, characteristic
4#20244340C211yesmaximal, characteristic, complemented
5#20244129C211yesmaximal, characteristic, complemented
6#20244114C211yesmaximal, characteristic, complemented
7#20244083C211yesmaximal, characteristic
8#20244069C211yesmaximal, characteristic
9#20244014C211yesmaximal, characteristic
10#20244007C211yesmaximal, characteristic
11#20243972C211yesmaximal, characteristic, complemented
12#20243965C211yesmaximal, characteristic, complemented
13#20243395C211yesmaximal, characteristic
14#20243247C211yesmaximal, characteristic, complemented
15C47×#1949C211yesmaximal, characteristic, complemented
16C2×#13094024C2211yescharacteristic
17C2×C4×D188C2211yescharacteristic
18#15689414C418yescomplemented
19C2×#13093956C2211yescharacteristic
20C2×#13093956C2211yescharacteristic
21C2×C4⋊(C47⋊C8)C2211yescharacteristic
22C2×#13093946C2211yescharacteristic
23C2×#13093946C2211yescharacteristic
24C2×#13093944C2211yescharacteristic
25C2×#13093944C2211yescharacteristic
26C2×#13093934C2211yescharacteristic
27C2×C8×Dic47C2211yescharacteristic
28#15689027—21no—
29#15689027—21no—
30#15689024C2211yescharacteristic
31#15689024C2211yescharacteristic
32#15689019C2211yescharacteristic
33#15689019C2211yescharacteristic
34#15689019C2211yescharacteristic
35#15689019C2211yescharacteristic
36#15689014—21no—
37#15689014—21no—
38#15689008—21no—
39#15689008—21no—
40#15688985—21no—
41#15688982—21no—
42C4⋊C4×Dic47C2211yescharacteristic
43C4⋊C4×Dic47C2211yescharacteristic
44#15688969C2211yescharacteristic
45#15688885C2211yescharacteristic
46#15688885C2211yescharacteristic
47#15688885C2211yescharacteristic
48#15688885C2211yescharacteristic
49#15688885C2211yescharacteristic
50#15688885C2211yescharacteristic
51#15688877C2211yescharacteristic
52#15688877C2211yescharacteristic
53C2×#107×C47C2211yescharacteristic
54C2×#106×C47C2211yescharacteristic
55C2×C4⋊C8×C47C2211yescharacteristic
56C47×#382C2211yescharacteristic
57C4×C4⋊C4×C47C2211yescharacteristic
58C47×#340C2211yescharacteristic
59C47×#340C2211yescharacteristic
60C22×Dic94—21no—
61C22×D188C2311yescharacteristic
62C22×C4×D47—21no—
63C22×C47⋊C8C2311yescharacteristic
64C22×C47⋊C8C2311yescharacteristic
65#13094024—22no—
66#13094024C2×C414yes—
67C4×D188—22no—
68C4×D188C2×C414yes—
69C2×C4⋊Dic47—21no—
70C2×C4⋊Dic47C2311yescharacteristic
71C2×C4⋊Dic47C2311yescharacteristic
72C2×C4⋊Dic47C2311yescharacteristic
73C2×Dic47⋊C4—21no—
74C2×Dic47⋊C4—21no—
75C2×Dic47⋊C4—21no—
76C2×Dic47⋊C4—21no—
77C2×(C2×D47)⋊C4—21no—
78C2×(C2×D47)⋊C4—21no—
79C2×(C2×D47)⋊C4—21no—
80C2×(C2×D47)⋊C4—21no—
81C2×(C2×D47)⋊C4—21no—
82C2×C4×Dic47—21no—
83C2×C4×Dic47—21no—
84C2×C4×Dic47C2311yescharacteristic
85C2×C4×Dic47C2311yescharacteristic
86#13093956C2×C414yes—
87#13093956C2×C414yes—
88C4⋊(C47⋊C8)—22no—
89#13093946C2×C414yes—
90#13093946C2×C414yes—
91#13093944—41no—
92#13093944—41no—
93#13093934—41no—
94C8×Dic47—41no—
95C47⋊#95—21no—
96C47⋊#95—21no—
97C47⋊#95—21no—
98C47⋊#95—21no—
99C47⋊#95—21no—
100C47⋊#95—21no—
101C47⋊#95D411yescharacteristic
102C47⋊#95D411yescharacteristic
103C47⋊#95D411yescharacteristic
104C47⋊#95D411yescharacteristic
105C22×C8×C47C2311yescharacteristic
106C22×C8×C47C2311yescharacteristic
107C2×C4⋊C4×C47C2311yescharacteristic
108C2×C4⋊C4×C47C2311yescharacteristic
109C2×C4⋊C4×C47C2311yescharacteristic
110C2×C4⋊C4×C47C2311yescharacteristic
111C2×C42×C47—21no—
112C2×C42×C47C2311yescharacteristic
113#107×C47—41no—
114#106×C47—41no—
115C4⋊C8×C47C2×C414yes—
116#95×C47—21no—
117#95×C47—21no—
118#95×C47—21no—
119C23×D47—21no—
120C22×C4×C47—21no—
121C22×C4×C47—21no—
122C22×C4×C47—21no—
123C22×C4×C47—21no—
124C22×C4×C47C2×D411yescharacteristic
125C22×C4×C47C2×D411yescharacteristic
126C22×C4×C47C2411yescharacteristic
127C22×Dic47—21no—
128C22×Dic47—21no—
129C22×Dic47—21no—
130C22×Dic47C2×D411yescharacteristic
131C22×Dic47C2×D411yescharacteristic
132C22×Dic47C2×D411yescharacteristic
133C22×Dic47C2×D411yescharacteristic
134C2×D188—21no—
135C2×D188—21no—
136C2×D188—21no—
137C2×D188—21no—
138C2×D188—21no—
139C2×D188C22×C412yes—
140C2×C4×D47—24no—
141C2×C4×D47—81no—
142C2×C4×D47—81no—
143C2×Dic94—24no—
144C2×Dic94—81no—
145C2×Dic94—81no—
146C2×C8×C47—21no—
147C2×C8×C47—21no—
148C2×C8×C47—21no—
149C2×C8×C47—21no—
150C2×C8×C47C22×C412yes—
151C2×C8×C47C22×C412yes—
152C4⋊C4×C47—21no—
153C4⋊C4×C47—21no—
154C4⋊C4×C47—21no—
155C4⋊C4×C47—21no—
156C4⋊C4×C47—21no—
157C4⋊C4×C47—21no—
158C4⋊C4×C47—21no—
159C4⋊C4×C47—21no—
160C42×C47—81no—
161C42×C47C22×C412yes—
162C42×C47#4112yes—
163(C2×D47)⋊C4—24no—
164(C2×D47)⋊C4—81no—
165(C2×D47)⋊C4—81no—
166(C2×D47)⋊C4—81no—
167(C2×D47)⋊C4—81no—
168C4⋊Dic47—24no—
169C4⋊Dic47—21no—
170C4⋊Dic47—21no—
171C4⋊Dic47—21no—
172C4⋊Dic47C22×C412yes—
173C4⋊Dic47C22×C412yes—
174C4⋊Dic47C22×C412yes—
175Dic47⋊C4—81no—
176Dic47⋊C4—81no—
177Dic47⋊C4—81no—
178Dic47⋊C4—81no—
179C4×Dic47—21no—
180C4×Dic47—21no—
181C4×Dic47—21no—
182C4×Dic47—21no—
183C4×Dic47—81no—
184C4×Dic47—81no—
185C2×C47⋊C8—21no—
186C2×C47⋊C8—21no—
187C2×C47⋊C8—21no—
188C2×C47⋊C8—21no—
189C2×C47⋊C8—21no—
190C2×C47⋊C8—21no—
191C23×C47C22×D411yescharacteristic
192C22×D47—22no—
193C22×D47—41no—
194C22×D47—41no—
195C22×D47—41no—
196C22×D47—41no—
197C22×D47—41no—
198C22×D47—41no—
199C2×C4×C47—21no—
200C2×C4×C47—21no—
201C2×C4×C47—21no—
202C2×C4×C47—21no—
203C2×C4×C47—41no—
204C2×C4×C47—41no—
205C2×C4×C47—41no—
206C2×C4×C47—41no—
207C2×C4×C47—41no—
208C2×C4×C47—41no—
209C2×C4×C47—41no—
210C2×C4×C47—41no—
211C2×C4×C47—41no—
212C2×C4×C47—41no—
213C2×C4×C47—41no—
214C2×C4×C47—41no—
215C2×C4×C47C4×D412yes—
216C2×C4×C47C4×D412yes—
217C2×C4×C47C23×C411yescharacteristic
218C2×C4×C47C2×#4111yescharacteristic
219C2×C4×C47H(5,𝔽2)11yescharacteristic
220C2×C4×C472−1+411yescharacteristic
221C2×C4×C472−1+411yescharacteristic
222C2×C4×C472−1+411yescharacteristic
223C2×Dic47—22no—
224C2×Dic47—22no—
225C2×Dic47—22no—
226C2×Dic47—21no—
227C2×Dic47—21no—
228C2×Dic47—21no—
229C2×Dic47—21no—
230C2×Dic47—21no—
231C2×Dic47—21no—
232C2×Dic47—21no—
233C2×Dic47—21no—
234C2×Dic47—21no—
235C2×Dic47—21no—
236C2×Dic47—21no—
237C2×Dic47—21no—
238C2×Dic47—41no—
239C2×Dic47—41no—
240C2×Dic47—41no—
241C2×Dic47—41no—
242C2×Dic47—41no—
243C2×Dic47—41no—
244D188—22no—
245D188—22no—
246D188—41no—
247D188—41no—
248C4×D47—84no—
249Dic94—84no—
250C8×C47—22no—
251C8×C47—22no—
252C47⋊C8—41no—
253C47⋊C8—41no—
254#22123—471nomaximal, Sylow
255C22×C47C2×C4×D411yescharacteristic
256C22×C47#53611yescharacteristic
257C22×C47#53711yescharacteristic
258C22×C47#57611yescharacteristic
259C22×C47#57711yescharacteristic
260C22×C47#57711yescharacteristic
261C22×C47#57711yescharacteristic
262C2×D47—42no—
263C2×D47—42no—
264C2×D47—42no—
265C2×D47—81no—
266C2×D47—81no—
267C2×D47—81no—
268C2×D47—81no—
269C4×C47—22no—
270C4×C47—22no—
271C4×C47—81no—
272C4×C47—81no—
273C4×C47—81no—
274C4×C47—81no—
275C4×C47#51911yescharacteristic
276C4×C47#52011yescharacteristic
277C4×C47#54111yescharacteristic
278C4×C47#54211yescharacteristic
279Dic47—42no—
280Dic47—42no—
281Dic47—42no—
282Dic47—41no—
283Dic47—41no—
284Dic47—41no—
285Dic47—41no—
286C2×#476—471no—
287#2331—471no—
288#2319—471no—
289#1988—471no—
290#1949D4711yescharacteristic, complemented
291#1944—471no—
292#1941—471no—
293#1936—471no—
294#1934—471no—
295#1926—471no—
296#1925—471no—
297#1920—471no—
298#1919—471no—
299#1804—471no—
300#1768—471no—
301C2×C47#232311yescharacteristic
302C2×C47#294611yescharacteristic
303C2×C47#294911yescharacteristic
304C2×C47#324111yescharacteristic
305C2×C47#324411yescharacteristic
306C2×C47#325711yescharacteristic
307C2×C47#326311yescharacteristic
308D47—82no—
309C2×C4⋊Q8—471no—
310C2×C4×D4—471no—
311#476—478no—
312C2×#107C2×D4711yescharacteristic
313C2×#106—471no—
314C2×#106—471no—
315C2×#106C2×D4711yescharacteristic
316C2×C4⋊C8—471no—
317C2×C4⋊C8C2×D4711yescharacteristic
318C2×#102—471no—
319C2×#102—471no—
320C2×#102—471no—
321C2×#102—471no—
322C2×#97—471no—
323C2×C4×C8—471no—
324#391—941no—
325#391—941no—
326#390—471no—
327#390—471no—
328#389—471no—
329#389—471no—
330#389—471no—
331#389—471no—
332#388—941no—
333#388—941no—
334#387—941no—
335#387—941no—
336#382C2×D4711yescharacteristic
337#381—941no—
338#380—941no—
339C4×C4⋊C4—471no—
340C4×C4⋊C4—471no—
341C4×C4⋊C4C2×D4711yescharacteristic
342#376—471no—
343#340—471no—
344#340—471no—
345#340—471no—
346#340—471no—
347#340—471no—
348#340—471no—
349#340C2×D4711yescharacteristic
350#340C2×D4711yescharacteristic
351#336—471no—
352#336—471no—
353C47#2212311yescharacteristic, complemented, Sylow
354C22×Q8—941no—
355C22×D4—471no—
356C23×C4—941no—
357C22×C8—471no—
358C22×C8—471no—
359C22×C8C22×D4711yescharacteristic
360C22×C8C22×D4711yescharacteristic
361C4⋊Q8—474no—
362C4⋊Q8—942no—
363C4×D4—474no—
364C4×D4—942no—
365C2×C4⋊C4—471no—
366C2×C4⋊C4—471no—
367C2×C4⋊C4—471no—
368C2×C4⋊C4—941no—
369C2×C4⋊C4—941no—
370C2×C4⋊C4—941no—
371C2×C4⋊C4—941no—
372C2×C4⋊C4—941no—
373C2×C4⋊C4C22×D4711yescharacteristic
374C2×C4⋊C4C22×D4711yescharacteristic
375C2×C4⋊C4C22×D4711yescharacteristic
376C2×C4⋊C4C22×D4711yescharacteristic
377C2×C22⋊C4—941no—
378C2×C22⋊C4—941no—
379C2×C22⋊C4—941no—
380C2×C22⋊C4—941no—
381C2×C22⋊C4—941no—
382C2×C42—21no—
383C2×C42—471no—
384C2×C42—471no—
385C2×C42—941no—
386C2×C42—941no—
387C2×C42C22×D4711yescharacteristic
388#107—41no—
389#106—41no—
390#106—474no—
391#106—474no—
392C4⋊C8—942no—
393C4⋊C8C4×D4714yes—
394#102—474no—
395#102—474no—
396#102—1881no—
397#102—1881no—
398#97—1881no—
399C4×C8—1881no—
400#95—21no—
401#95—21no—
402#95—21no—
403#95—471no—
404#95—471no—
405#95—471no—
406#95—471no—
407#95—941no—
408#95—941no—
409#95—941no—
410#95—941no—
411#95—941no—
412#95—941no—
413C24—941no—
414C2×Q8—944no—
415C2×Q8—3761no—
416C2×Q8—3761no—
417C2×D4—472no—
418C2×D4—941no—
419C2×D4—941no—
420C2×D4—941no—
421C2×D4—941no—
422C2×D4—941no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—471no—
428C22×C4—471no—
429C22×C4—471no—
430C22×C4—471no—
431C22×C4—941no—
432C22×C4—941no—
433C22×C4—941no—
434C22×C4—944no—
435C22×C4—3761no—
436C22×C4—3761no—
437C22×C4D4×D4711yescharacteristic
438C22×C4D4×D4711yescharacteristic
439C22×C4C23×D4711yescharacteristic
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—941no—
445C2×C8—941no—
446C2×C8—941no—
447C2×C8—941no—
448C2×C8—941no—
449C2×C8—941no—
450C2×C8C2×C4×D4712yes—
451C2×C8C2×C4×D4712yes—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—472no—
461C4⋊C4—472no—
462C4⋊C4—472no—
463C4⋊C4—944no—
464C4⋊C4—941no—
465C4⋊C4—941no—
466C4⋊C4—941no—
467C4⋊C4—3761no—
468C4⋊C4—3761no—
469C4⋊C4—3761no—
470C4⋊C4—3761no—
471C22⋊C4—944no—
472C22⋊C4—3761no—
473C22⋊C4—3761no—
474C22⋊C4—3761no—
475C22⋊C4—3761no—
476C42—81no—
477C42—941no—
478C42—941no—
479C42—941no—
480C42—941no—
481C42—3761no—
482C42—3761no—
483C42C2×C4×D4712yes—
484C42#1062535412yes—
485C23—942no—
486C23—1881no—
487C23—1881no—
488C23—1881no—
489C23—1881no—
490C23—1881no—
491C23—1881no—
492C23C2×D4×D4711yescharacteristic
493Q8—3764no—
494D4—942no—
495D4—942no—
496D4—1881no—
497D4—1881no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—942no—
515C2×C4—942no—
516C2×C4—942no—
517C2×C4—941no—
518C2×C4—941no—
519C2×C4—941no—
520C2×C4—941no—
521C2×C4—941no—
522C2×C4—941no—
523C2×C4—941no—
524C2×C4—941no—
525C2×C4—941no—
526C2×C4—941no—
527C2×C4—941no—
528C2×C4—941no—
529C2×C4—3764no—
530C2×C4—1881no—
531C2×C4—1881no—
532C2×C4—1881no—
533C2×C4—1881no—
534C2×C4—1881no—
535C2×C4—1881no—
536C2×C4#1309398612yes—
537C2×C4#1309398612yes—
538C2×C4C22×C4×D4711yescharacteristic
539C2×C4C2×#1062535411yescharacteristic
540C2×C4#1309406911yescharacteristic
541C2×C4#1309407111yescharacteristic
542C2×C4#1309407111yescharacteristic
543C2×C4#1309407211yescharacteristic
544C8—22no—
545C8—22no—
546C8—1881no—
547C8—1881no—
548C22—3761no—
549C22—3761no—
550C22—3761no—
551C22—3761no—
552C22—1882no—
553C22—1882no—
554C22—1882no—
555C22C2×#1309398611yescharacteristic
556C22#1568969611yescharacteristic
557C22#1568970211yescharacteristic
558C22#1568992711yescharacteristic
559C22#1568993011yescharacteristic
560C22#1568993011yescharacteristic
561C22#1568993211yescharacteristic
562C4—22no—
563C4—22no—
564C4—81no—
565C4—81no—
566C4—81no—
567C4—81no—
568C4—1882no—
569C4—1882no—
570C4—1881no—
571C4—1882no—
572C4—1881no—
573C4—1881no—
574C4—1881no—
575C4#1568959911yescharacteristic
576C4#1568960211yescharacteristic
577C4#1568973611yescharacteristic
578C4#1568974711yescharacteristic
579C2—3762no—
580C2#2024606811yescharacteristic
581C2#2025221811yescharacteristic
582C2#2025224611yescharacteristic
583C2#2025464611yescharacteristic
584C2#2025469311yescharacteristic
585C2#2025480411yescharacteristic
586C2#2025485611yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1949C2
socleC23×C47C22×D4
prime radicalC22×D188C23
cosocleC22×C4×C47C24
hypercenter#1949D47
hypocenterC47#22123
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1949

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #20246134C2semidirect
4. #20244340C2semidirect
5. #20244129C2semidirect
6. #20244114C2semidirect
11. #20243972C2semidirect
12. #20243965C2semidirect
14. #20243247C2semidirect
15. C47×#1949C2semidirect
18. #15689414C4semidirect
290. #1949D47semidirect
353. C47#22123semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#156894142
#202461342
#202460262
#202443402
#202441292
#202441142
#202440832
#202440692
#202440142
#202440072
#202439722
#202439652
#202433952
#202432472
C47×#19492
#2212347